Uncertainty Quantification in Fatigue Crack Growth Prognosis
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Published
Jan 1, 2011
Shankar Sankararaman
You Ling
Christopher Shantz
Sankaran Mahadevan
Abstract
This paper presents a methodology to quantify the uncertainty in fatigue crack growth prognosis, applied to structures with complicated geometry and subjected to variable amplitude multi-axial loading. Finite element analysis is used to address the complicated geometry and calculate the stress intensity factors. Multi-modal stress intensity factors due to multi-axial loading are combined to calculate an equivalent stress intensity factor using a characteristic plane approach. Crack growth under variable amplitude loading is modeled using a modified Paris law that includes retardation effects. During cycle-by-cycle integration of the crack growth law, a Gaussian process surrogate model is used to replace the expensive finite element analysis. The effect of different types of uncertainty – physical variability, data uncertainty and modeling errors – on crack growth prediction is investigated. The various sources of uncertainty include, but not limited to, variability in loading conditions, material parameters, experimental data, model uncertainty, etc. Three different types of modeling errors – crack growth model error, discretization error and surrogate model error – are included in analysis. The different types of uncertainty are incorporated into the crack growth prediction methodology to predict the probability distribution of crack size as a function of number of load cycles. The proposed method is illustrated using an application problem, surface cracking in a cylindrical structure
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Keywords
model-based methods, Fatigue Prognosis, Uncertainty Quantification, Natural Variability, Data Uncertainty, Model Error, Fracture Mechanics, Crack Growth
References
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J.P. Gallagher, A.P. Berens, and R.M. Engle Jr. (1984). USAF damage tolerant design handbook: guidelines for the analysis and design of damage tolerant aircraft structures. Final report. 1984.
A. Merati, and G. Eastaugh.(2007). Determination of fatigue related discontinuity state of 7000 series of aerospace aluminum alloys. Eng Failure Anal 2007; 14(4):673–85.
J.N. Yang.(1980) Distribution of equivalent initial flaw size. In: Proceedings of the annual reliability and maintainability symposium. San Francisco (CA): 1980.
P. White, L. Molent, and S. Barter (2005). Interpreting fatigue test results using a probabilistic fracture approach. Int J Fatigue 2005;27(7):752–67.
L. Molent, Q. Sun, and A. Green (2006) Charcterisation of equivalent initial flaw sizes in 7050 aluminium alloy. J Fatigue Fract Eng Mater Struct 2006;29:916–37.
PMGP. Moreira, PFP. de Matos, and PMST. de Castro (2005) Fatigue striation spacing and equivalent initial flaw size in Al 2024-T3 riveted specimens. Theor Appl FractMech 2005;43(1):89–99.
S.A. Fawaz (2000). Equivalent initial flaw size testing and analysis.
Y. Liu and S. Mahadevan (2008). Probabilistic fatigue life prediction using an equivalent initial flaw size distributio., International Journal of Fatigue, Volume 31, Issue 3, March 2009, Pages 476-487, ISSN 0142-1123, DOI: 10.1016/j.ijfatigue.2008.06.005.
H. Kitagawa, and S. Takahashi (1976). Applicability of fracture mechanics to vary small cracks or cracks in early stage. In: Proceedings of the 2nd international conference on mechanical behavior of materials. USA (OH): ASM International.
M.H. El Haddad, T.H. Topper, and K.N. Smith (1979). Prediction of nonpropagating cracks. Eng Fract Mech 1979;11:573–84.
F. Hemez (2005) Uncertainty Quantification and the Verification and Validation of Computational Models. Chapter in Damage Prognosis for for Aerospace, Civil, and Mechanical Systems. Edited by D. J. Inman, C.R. Farrar, V. Lopes, and V. Steffen. 2005. ISBN 0-470-86907-0. John Wiley & Sons Ltd.
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D. Chelidze and J.P. Cusumano (2004) A Dynamical Systems Approach to Systems Prognosis. J. Vib. Acoust. 126 (2). 2004. DOI. 10.1115/1.1640638.
B. Saha and K. Goebel (2008) Uncertainty Management for Diagnostics and Prognostics of Batteries using Bayesian Techniques. 2008. Proceedings of the IEEE Aerospace Conference 2008. Big Sky, Montana. Mat 1 – Mar 8, 2008.
K. Medjaher, J. Y. Moya, and N. Zerhouni (2009) Failure Prognostic by Using Dynamic Bayes Networks. In the Proceedings of the 2nd IFAC Workshop on Dependable Control of Discrete Systems. DCDS 2009. July 1 – 8, Bari, Italy.
S.W. Doebling, and F.M. Hemez (2001) Overview of Uncertainty Assessment for Structural Health Monitoring. In the Proceedings of the 3rd International Workshop on Structural Health Monitoring, September 17-19, 2001, Stanford University, Stanford, California.
F.M. Hemez, A.N. Roberson, and A.C. Rutherford (2003) Uncertainty Quantification and Model Validation for Damage Prognosis. In the Proceedings of the 4th International Workshop on Structural Health Monitoring, Stanford University, Stanford, California, September 15-17, 2003
C.R. Farrar, G. Park, F.M. Hemez, T.B. Tippetts, H. Sohn, J. Wait, D.W. Allen, and B.R. Nadler (2004). Damage Detection and Prediction for Composite Plates. J. of The Minerals, Metals and Materials Society, November 2004.
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R. Patrick, M.E. Orchard, B. Zhang, M.D. Koelemay, G.J. Kacprzynski, A.A. Ferri, and G.J. Vachtsevanos (2007) An integrated approach to helicopter planetary gear fault diagnosis and failure prognosis. Autotestcon, 2007 IEEE , vol., no., pp.547-552, 17-20 Sept. 2007.
S. Gupta, and A. Ray (2007) Real-time fatigue life estimation in mechanical structures. Meas. Sci. Technol. 18 (2007) 1947–1957. doi:10.1088/0957-0233/18/7/022.
S.G. Pierce, K. Worden, and A. Bezazi (2008) Uncertainty analysis of a neural network used for fatigue lifetime prediction. Mechanical Systems and Signal Processing, Volume 22, Issue 6, Special Issue: Mechatronics, August 2008, Pages 1395-1411, ISSN 0888-3270, DOI: 10.1016/j.ymssp.2007.12.004.
J.M. Papazian, E.L. Anagnostou, S.J. Engel, D. Hoitsma, J. Madsen, R.P. Silberstein, G. Welsh, and J.B. Whiteside (2009) A structural integrity prognosis system. Engineering Fracture Mechanics, Volume 76, Issue 5, Material Damage Prognosis and Life-Cycle Engineering, March 2009, Pages 620-632, ISSN 0013-7944, DOI: 10.1016/j.engfracmech.2008.09.007.
Y. Liu and S. Mahadevan (2005) Multiaxial high-cycle fatigue criterion and life prediction for metals. International Journal of Fatigue, 2005. 27(7): p. 790-80
C. Rasmussen (1996) Evaluation of Gaussian processes and other methods for non-linear regression. PhD thesis, University of Toronto, 1996
T.J. Santner, B.J. Williams, and W.I. Noltz. (2003). The Design and Analysis of Computer Experiments. Springer-Verlag, New York, 2003.
Bichon, B., Eldred, M., Swiler, L., Mahadevan, S., and McFarland, J. (2008). Efficient Global Reliability Analysis for Nonlinear Implicit Performance Functions. AIAA Journal 2008. 0001-1452 vol.46 No.10 (2459-2468). doi: 10.2514/1.34321.
J. Sacks, S. B. Schiller, and W. Welch (1989) “Design for Computer Experiments,” Technometrics, Vol. 31, No. 1, 1989, pp. 41–47. doi:10.2307/1270363.
J. McFarland (2008) Uncertainty analysis for computer simulations through validation and calibration. Ph D. Dissertation, Vanderbilt University, 2008
S.A. Richards (1997) Completed Richardson extrapolation in space and time. Comm Numer Methods Eng 1997;13:558–73.
R. Rebba (2002) Computational model validation under uncertainty.Master’s thesis. Nashville, TN: Vanderbilt University.
S.A. Barter, P.K. Sharp, G. Holden, and G. Clark (2002) Initiation and early growth of fatigue cracks in an aerospace aluminum alloy, Fatigue Fract. Eng. Mater. 25 (2002) (2), pp. 111–125.
A. Makeev, Y. Nikishkov, and E. Armanios (2007) A concept for quantifying equivalent initial flaw size distribution in fracture mechanics based life prediction models, Int J Fatigue (2006), Vol. 29, No. 1, Jan. 2007.
R. Cross, A. Makeev, and E. Armanios (2007) Simultaneous uncertainty quantification of fracture mechanics based life prediction model parameters, Int J Fatigue, Vol. 29, No. 8, Aug. 2007.
R. Rebba, S. Mahadevan, and S. Huang (2006) Validation and error estimation of computational models, Reliability Engineering & System Safety, Volume 91, Issues 10-11, The Fourth International Conference on Sensitivity Analysis of Model Output (SAMO 2004) - SAMO 2004, October-November 2006, Pages 1390-1397, ISSN0951-8320, DOI 10.1016/j.ress.2005.11.035
B. Efron, and R.J. Tibshirani (1993) An Introduction to the Bootstrap. Monographs on Applied Statistics and Probability 57. Chapam and Hall/CRC. 1993.
B. Efron (1979) Bootstrap Methods: Another look at the Jackknife. The annals of statistics. Vol. 7. No. 1. pp 1-26. 1979.
M. McDonald, K. Zaman, and S. Mahadevan (2009) Representation and First-Order Approximations for Propagation of Aleatory and Distribution Parameter Uncertainty. In the Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 4 - 7 May 2009, Palm Springs, California.
A. Haldar, and S. Mahadevan (2000) Probability, Reliability and Statistical Methods in Engineering Design, Wiley, New York, 2000.
ANSYS (2007) ANSYS theory reference, release 11.0. ANSYS Inc., 2007.
Y. Liu, L. Liu, and S. Mahadevan (2007) Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM. Engineering Fracture Mechanics, Vol 74, pgs 2659-2674, 2007.
G.H. Besterfield, W.K. Liu, A.M. Lawrence, and T. Belytschko (1991) Fatigue crack growth reliability by probabilistic finite elements, Computer Methods in Applied Mechanics and Engineering, Volume 86, Issue 3.
G.A.F. Seber, and C.J. Wild (1989) Nonlinear Regression. New York: John Wiley and Sons.
M. Orchard, G. Kacprzynski, K. Goebel, B. Saha, and G. Vachtsevanos (2008) Advances in Uncertainty Representation and Management for Particle Filtering Applied to Prognostics. In the Proceedings of the 1st Prognostics and Health Management (PHM) Conference, Denver, CO. Oct 6-9, 2008.
P.S. Song, B.C. Sheu, and L. Chang (2001) A modified wheeler model to improve predictions of crack growth following a single overload. JSME Int J Series A 2001;44(1):117–22.
B.K.C. Yuen, and F. Taheri (2006) Proposed modifications to the Wheeler retardation model for multiple overloading fatigue life prediction. International Journal of Fatigue, Volume 28, Issue 12, December 2006, Pages 1803-1819, ISSN 0142-1123, DOI: 10.1016/j.ijfatigue.2005.12.007.
B.C. Sheu, P.S. Song, and S. Hwang (1995) Shaping exponent in wheeler model under a single overload. Eng Fract Mech 1995;51(1): 135–43.
J. Schijve (1976) Observations on the Predictions of Fatigue Crack Growth Prediction under Variable Amplitude loading. ASTM STP 595, 1976, pp. 3-23.
A.H. Noroozi, G. Glinka, S. Lambert (2008) Prediction of fatigue crack growth under constant amplitude loading and a single overload based on elasto-plastic crack tip stresses and strains, Engineering Fracture Mechanics, Volume 75, Issue 2, January 2008, Pages 188-206.
J.P. Gallagher, A.P. Berens, and R.M. Engle Jr. (1984). USAF damage tolerant design handbook: guidelines for the analysis and design of damage tolerant aircraft structures. Final report. 1984.
A. Merati, and G. Eastaugh.(2007). Determination of fatigue related discontinuity state of 7000 series of aerospace aluminum alloys. Eng Failure Anal 2007; 14(4):673–85.
J.N. Yang.(1980) Distribution of equivalent initial flaw size. In: Proceedings of the annual reliability and maintainability symposium. San Francisco (CA): 1980.
P. White, L. Molent, and S. Barter (2005). Interpreting fatigue test results using a probabilistic fracture approach. Int J Fatigue 2005;27(7):752–67.
L. Molent, Q. Sun, and A. Green (2006) Charcterisation of equivalent initial flaw sizes in 7050 aluminium alloy. J Fatigue Fract Eng Mater Struct 2006;29:916–37.
PMGP. Moreira, PFP. de Matos, and PMST. de Castro (2005) Fatigue striation spacing and equivalent initial flaw size in Al 2024-T3 riveted specimens. Theor Appl FractMech 2005;43(1):89–99.
S.A. Fawaz (2000). Equivalent initial flaw size testing and analysis.
Y. Liu and S. Mahadevan (2008). Probabilistic fatigue life prediction using an equivalent initial flaw size distributio., International Journal of Fatigue, Volume 31, Issue 3, March 2009, Pages 476-487, ISSN 0142-1123, DOI: 10.1016/j.ijfatigue.2008.06.005.
H. Kitagawa, and S. Takahashi (1976). Applicability of fracture mechanics to vary small cracks or cracks in early stage. In: Proceedings of the 2nd international conference on mechanical behavior of materials. USA (OH): ASM International.
M.H. El Haddad, T.H. Topper, and K.N. Smith (1979). Prediction of nonpropagating cracks. Eng Fract Mech 1979;11:573–84.
F. Hemez (2005) Uncertainty Quantification and the Verification and Validation of Computational Models. Chapter in Damage Prognosis for for Aerospace, Civil, and Mechanical Systems. Edited by D. J. Inman, C.R. Farrar, V. Lopes, and V. Steffen. 2005. ISBN 0-470-86907-0. John Wiley & Sons Ltd.
D. J. Inman, C.R. Farrar, V. Lopes, and V. Steffen (2005) Damage Prognosis for Aerospace, Civil, and Mechanical Systems. 2005. ISBN 0-470-86907-0. John Wiley & Sons Ltd.
D. Chelidze and J.P. Cusumano (2004) A Dynamical Systems Approach to Systems Prognosis. J. Vib. Acoust. 126 (2). 2004. DOI. 10.1115/1.1640638.
B. Saha and K. Goebel (2008) Uncertainty Management for Diagnostics and Prognostics of Batteries using Bayesian Techniques. 2008. Proceedings of the IEEE Aerospace Conference 2008. Big Sky, Montana. Mat 1 – Mar 8, 2008.
K. Medjaher, J. Y. Moya, and N. Zerhouni (2009) Failure Prognostic by Using Dynamic Bayes Networks. In the Proceedings of the 2nd IFAC Workshop on Dependable Control of Discrete Systems. DCDS 2009. July 1 – 8, Bari, Italy.
S.W. Doebling, and F.M. Hemez (2001) Overview of Uncertainty Assessment for Structural Health Monitoring. In the Proceedings of the 3rd International Workshop on Structural Health Monitoring, September 17-19, 2001, Stanford University, Stanford, California.
F.M. Hemez, A.N. Roberson, and A.C. Rutherford (2003) Uncertainty Quantification and Model Validation for Damage Prognosis. In the Proceedings of the 4th International Workshop on Structural Health Monitoring, Stanford University, Stanford, California, September 15-17, 2003
C.R. Farrar, G. Park, F.M. Hemez, T.B. Tippetts, H. Sohn, J. Wait, D.W. Allen, and B.R. Nadler (2004). Damage Detection and Prediction for Composite Plates. J. of The Minerals, Metals and Materials Society, November 2004.
C.R. Farrar, and N.A.J. Lieven (2006) Damage Prognosis The Future of Structural Health Monitoring. Phil. Trans. R. Soc. 365, 623–632 doi:10.1098/rsta.2006.1927. Published online 12 December 2006.
R. Patrick, M.E. Orchard, B. Zhang, M.D. Koelemay, G.J. Kacprzynski, A.A. Ferri, and G.J. Vachtsevanos (2007) An integrated approach to helicopter planetary gear fault diagnosis and failure prognosis. Autotestcon, 2007 IEEE , vol., no., pp.547-552, 17-20 Sept. 2007.
S. Gupta, and A. Ray (2007) Real-time fatigue life estimation in mechanical structures. Meas. Sci. Technol. 18 (2007) 1947–1957. doi:10.1088/0957-0233/18/7/022.
S.G. Pierce, K. Worden, and A. Bezazi (2008) Uncertainty analysis of a neural network used for fatigue lifetime prediction. Mechanical Systems and Signal Processing, Volume 22, Issue 6, Special Issue: Mechatronics, August 2008, Pages 1395-1411, ISSN 0888-3270, DOI: 10.1016/j.ymssp.2007.12.004.
J.M. Papazian, E.L. Anagnostou, S.J. Engel, D. Hoitsma, J. Madsen, R.P. Silberstein, G. Welsh, and J.B. Whiteside (2009) A structural integrity prognosis system. Engineering Fracture Mechanics, Volume 76, Issue 5, Material Damage Prognosis and Life-Cycle Engineering, March 2009, Pages 620-632, ISSN 0013-7944, DOI: 10.1016/j.engfracmech.2008.09.007.
Y. Liu and S. Mahadevan (2005) Multiaxial high-cycle fatigue criterion and life prediction for metals. International Journal of Fatigue, 2005. 27(7): p. 790-80
C. Rasmussen (1996) Evaluation of Gaussian processes and other methods for non-linear regression. PhD thesis, University of Toronto, 1996
T.J. Santner, B.J. Williams, and W.I. Noltz. (2003). The Design and Analysis of Computer Experiments. Springer-Verlag, New York, 2003.
Bichon, B., Eldred, M., Swiler, L., Mahadevan, S., and McFarland, J. (2008). Efficient Global Reliability Analysis for Nonlinear Implicit Performance Functions. AIAA Journal 2008. 0001-1452 vol.46 No.10 (2459-2468). doi: 10.2514/1.34321.
J. Sacks, S. B. Schiller, and W. Welch (1989) “Design for Computer Experiments,” Technometrics, Vol. 31, No. 1, 1989, pp. 41–47. doi:10.2307/1270363.
J. McFarland (2008) Uncertainty analysis for computer simulations through validation and calibration. Ph D. Dissertation, Vanderbilt University, 2008
S.A. Richards (1997) Completed Richardson extrapolation in space and time. Comm Numer Methods Eng 1997;13:558–73.
R. Rebba (2002) Computational model validation under uncertainty.Master’s thesis. Nashville, TN: Vanderbilt University.
S.A. Barter, P.K. Sharp, G. Holden, and G. Clark (2002) Initiation and early growth of fatigue cracks in an aerospace aluminum alloy, Fatigue Fract. Eng. Mater. 25 (2002) (2), pp. 111–125.
A. Makeev, Y. Nikishkov, and E. Armanios (2007) A concept for quantifying equivalent initial flaw size distribution in fracture mechanics based life prediction models, Int J Fatigue (2006), Vol. 29, No. 1, Jan. 2007.
R. Cross, A. Makeev, and E. Armanios (2007) Simultaneous uncertainty quantification of fracture mechanics based life prediction model parameters, Int J Fatigue, Vol. 29, No. 8, Aug. 2007.
R. Rebba, S. Mahadevan, and S. Huang (2006) Validation and error estimation of computational models, Reliability Engineering & System Safety, Volume 91, Issues 10-11, The Fourth International Conference on Sensitivity Analysis of Model Output (SAMO 2004) - SAMO 2004, October-November 2006, Pages 1390-1397, ISSN0951-8320, DOI 10.1016/j.ress.2005.11.035
B. Efron, and R.J. Tibshirani (1993) An Introduction to the Bootstrap. Monographs on Applied Statistics and Probability 57. Chapam and Hall/CRC. 1993.
B. Efron (1979) Bootstrap Methods: Another look at the Jackknife. The annals of statistics. Vol. 7. No. 1. pp 1-26. 1979.
M. McDonald, K. Zaman, and S. Mahadevan (2009) Representation and First-Order Approximations for Propagation of Aleatory and Distribution Parameter Uncertainty. In the Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 4 - 7 May 2009, Palm Springs, California.
A. Haldar, and S. Mahadevan (2000) Probability, Reliability and Statistical Methods in Engineering Design, Wiley, New York, 2000.
ANSYS (2007) ANSYS theory reference, release 11.0. ANSYS Inc., 2007.
Y. Liu, L. Liu, and S. Mahadevan (2007) Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM. Engineering Fracture Mechanics, Vol 74, pgs 2659-2674, 2007.
G.H. Besterfield, W.K. Liu, A.M. Lawrence, and T. Belytschko (1991) Fatigue crack growth reliability by probabilistic finite elements, Computer Methods in Applied Mechanics and Engineering, Volume 86, Issue 3.
G.A.F. Seber, and C.J. Wild (1989) Nonlinear Regression. New York: John Wiley and Sons.
M. Orchard, G. Kacprzynski, K. Goebel, B. Saha, and G. Vachtsevanos (2008) Advances in Uncertainty Representation and Management for Particle Filtering Applied to Prognostics. In the Proceedings of the 1st Prognostics and Health Management (PHM) Conference, Denver, CO. Oct 6-9, 2008.
P.S. Song, B.C. Sheu, and L. Chang (2001) A modified wheeler model to improve predictions of crack growth following a single overload. JSME Int J Series A 2001;44(1):117–22.
B.K.C. Yuen, and F. Taheri (2006) Proposed modifications to the Wheeler retardation model for multiple overloading fatigue life prediction. International Journal of Fatigue, Volume 28, Issue 12, December 2006, Pages 1803-1819, ISSN 0142-1123, DOI: 10.1016/j.ijfatigue.2005.12.007.
B.C. Sheu, P.S. Song, and S. Hwang (1995) Shaping exponent in wheeler model under a single overload. Eng Fract Mech 1995;51(1): 135–43.
J. Schijve (1976) Observations on the Predictions of Fatigue Crack Growth Prediction under Variable Amplitude loading. ASTM STP 595, 1976, pp. 3-23.
A.H. Noroozi, G. Glinka, S. Lambert (2008) Prediction of fatigue crack growth under constant amplitude loading and a single overload based on elasto-plastic crack tip stresses and strains, Engineering Fracture Mechanics, Volume 75, Issue 2, January 2008, Pages 188-206.
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